The theory of n-categories is currently under active consideration by a number of different research groups around the world. The history of the subject goes back a long way, on separate but interrelated tracks in algebraic topology, algebraic geometry and category theory. For a long time, the crucial definition of weakly associative higher category remained elusive, but now we have a plethora of different possibilities available. One of the next major problems in the subject will be to achieve a global comparison between these different approaches. Some work is starting to come out in this direction, but in the current state of the theory the various different approaches remain distinct. After the comparison is achieved, they will be seen as representing different facets of the theory, so it is important to continue working in all of these different directions.

The purpose of the present book is to concentrate on one of the methods of defining and working with higher categories, very closely based on the work of Graeme Segal in algebraic topology many years earlier. The notion of “Segal category,” which is a kind of category weakly enriched over simplicial sets, was considered by Schwänzl, Vogt and Dwyer, Kan and Smith. The application of this method to n-categories was introduced by Zouhair Tamsamani. It was then put into a strictly iterative form, with a general model category as input, by Regis Pellissier following a suggestion of André Hirschowitz.